$\newcommand{\indep}{\perp\!\!\!\perp}$ In discrete time, the Markov property is $$P[X_{n+1}\in A\mid X_n=s_n,X_{n-1}=s_{n-1}\dots ]=P[X_{n+1}\in A\mid X_n=s_n]$$ On the other hand, the "general Markov property" as given in Kallenberg's Foundations of Moderns Probability is the conditional independence $$X_u \indep _{X_t}{} \mathcal{F}_t$$ for any $t\leq u$.
What's bugging me is the freedom of $u$ and $t$ in the general case compared to $n+1$ and $n$ in the discrete time - in the general case one could take $u$ much larger than $t$, yet there is nothing like $$P[X_{m}\in A\mid X_n=s_n,X_{n-1}=s_{n-1}\dots ]=P[X_{m}\in A\mid X_n=s_n]$$ for $m\geq n$.
Why is the discrete time formulation as it is?
The version of the Markov property in discrete time that you stated in your post implies the more general property that, for every $k\geqslant1$, $$P[X_{n+k}\in A\mid X_n=s_n,X_{n-1}=s_{n-1},\ldots,X_0=s_0]=P[X_{n+k}\in A\mid X_n=s_n].$$ Equivalently, for every $m\geqslant n+1$, $$X_m \indep_{X_n}\mathcal{F}_n.$$